Given two graphs and their tree decompositions, computing the isomorphism respecting these tree decompositions is reducible to (1).

Given tree decomposition of only one graph, we can guess the tree decomposition of the other and guess the isomorphism (respecting the tree bags) and verify them using a non-deterministic auxiliary pushdown automata (a.k.a LogCFL).

Since tree decomposition of a graph can be computed in LogCFL, the above theorem follows.

One of the bottlenecks, finding a tree decomposition of bounded treewidth graphs in logspace, is resolved by [Elberfeld, Jakoby and Tantau’10]. The following seems to be another major bottleneck.

Given a graph and a decomposition , how fast can we verify that is a valid tree decomposition of ? The upper bound of LogDCFL (the deterministic version of LogCFL) is clear from the above mentioned results. Can this verification be done in logspace ?

The answer is frustratingly unknown. An even more frustrating realization I had today is that “it is not clear how to beat the LogDCFL upper bound for the more restricted path decomposition“. Even though the underlying tree in a path decomposition is just a path, verifying the connectivity conditions of a path decomposition does seem to require recursion. It is not clear how to avoid recursion.

I thought that logspace upper bound is possible. Now I am much less confident about logspace upper bound. I cannot waste more time on this.

The truth is “this is a cute problem“. I need to do something to take my mind off this problem and move on. Easy enough, except I need an idea.